4.2 Grand Ensemble: Open Systems
193
The grand partition function in this case will be given by
, μ) = 1 + 2e
β(I p +μ) ,
and the probability that a donor will ionize is thus
p i =
1
1 + 2e β(I p +μ) .
The chemical potential for an ideal gas of conduction electrons is
μ = −k B T ln
V z e −
int
N ce ce
,
with z e −
int = 2 due to the electron spin degeneracy for the spin1
2 electron, while ce
is the thermal de Broglie wavelength corresponding to the effective mass m ∗
ce of a
conduction electron.
The probability for the ionization of a donor atom is given by the ratio of the
number of conduction electrons, N ce , to the number of donor atoms, N d , i.e.,
p i =
N ce
N d
=
1
1 + 2e βI p (N ce 3
ce /2V )
=
1
1 + N ce 3
ce e I p /(k B T ) /V
,
from which N ce is obtained as
N ce =
N c e −I p /(k B T )
4
−1 +
1 +
8N d
N c
e I p /(k B T )
.
We shall plot N ce /N d vs. T // I , for T ranging from 0 to I , with I is given by
I ≡ I p /k B T . Thus N ce /N d takes the form
N ce
N d
=
N c
N d
e − I /T
4
−1 +
1 +
8N d
N c
e I /T
.
For N d = 10 23 m −3 and m ∗
ce = 0.45m e , we may express N c as
N c = 2
2πm ∗
ce k B T
h 2
3
2 = 4.613 × 10
22 T
3
2 m
−3 ,
from which we find that N c /(4N d ) = 0.1153T
3
2 . Moreover, using k B = 8.618 ×
10 −5 eV K
−1 , the characteristic temperature for this doped semiconductor has the